* Appendix A: Spherical Tensor Formalism
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* Appendix A: Spherical Tensor Formalism The geometrical aspects of the quantum-mechanical properties of molecular systems are described in terms of quantities familiar from angular momentum theory and the reader is referred to the text books existing on this subject. 1- 3 In this appendix we collect together the quantities and relations most fre quently used in calculations concerning the properties of solid hydrogen and other systems of interacting molecules. A.1 Spherical Harmonics The spherical harmonics Y,m(O,<jJ) are defined as the angular parts of the simultaneous eigenfunctions of the square and the :: component of the orbital angular momentum of a single particle, measured in units h (A.1) where m = I, I - 1, ... , -I and I = 0, 1,2, .... These functions are mu tually orthogonal and normalized on the unit sphere, (1m I I'm') == SOh S: Y;';" Y,'m,sine de d<jJ = ()ll'()mm' (A.2) The Y,m are determined uniquely by eqs. (A.1)-(A.2) up to arbitrary phase factors which we fix by choosing the so-called Condon and Shortley phases defined by the conditions that the nonvanishing matrix elements of the operators L, ± iLy and the values of the Y,o at 0 = °are real and positive (Lx ± iL)Y,m = [(I ± m + 1)(1 =+= m)]1'2y,.m±1 (A.3) and 21 + 1)1.12 Y,o(O,<jJ) = ( ~ (AA) 277 278 Appendix A • Spherical Tensor Formalism F or this choice of phases Y'tr,(8,c/J) = (-lrr;m(8,c/J) (A.5) where m == -m. Many expressions are simplified by using the Racah spherical harmonics defined as (A.6) which satisfy (A.7) where PI is the Legendre polynomial of degree I. For I = 0, ... , 4 and m > 0 these functions are given in Table A.1, and for m > 0 can be obtained from eg. (A.5). Hence explicit minus signs in the expressions for the elm and r;m occur only for positive odd values of m. The transformation law of the spherical harmonics under rotations of the coordinate frame is (A.S) m where 8,c/J and 8',c/J' are the polar angles of a given direction (unit vector) in the old and the new frame, respectively, and ~, /3, " are the Euler angles of the rotation carrying the old into the new frame. This rotation is effected by Table A.1 The Racah spherical harmonics up to I = 4. Coo = 1 C IO = cosO C 20 = t(3cos28 - 1) C 30 = lcos8(icos18 - 1) C40 = i(¥cos4 0 - lOcos28 + 1) C ll = -rtll/2sin!ie'4> C 21 = - (~)1!2sin8cosOe'4> C 31 = -i,/3sin8(5cos28 - Ije'4> C41 = -i,JSsin8cos8(1cos28 - l)e,4> C22 = t(~)1'2sin28e2i,p C 31 = HY/ 2sin18cos8e2',p C 41 = i(W 2 sin28(7cos28 - l)e2i 4> C 33 = -i.j5sin38e 3,,p 3 C43 = -iJSsin38cos!ie ',p C 44 = ~"isin 40e4,,p A.1 • Spherical Harmonics 279 first rotating the old frame x,y,.: into X 1,Y1,':1 = .: by a rotation through an angle oc around the z axis, then through an angle [3 around the )' 1 axis to produce a frame XhV2 = )'1' ':2 = .:' and finally through an angle ~' around the z' axis to produce the new frame x',y',.:'. The same transformation is obtained by rotations around the original axes, first through ~' around .:, then through [3 around y, and finally through :x around.: again. The rotatioll operator in the Hilbert space of a quantum-mechanical system with total angular momentum J. corresponding to the rotation R = (:x.[3.~') is and the rotatioll matrices are defined as D!",,(R) = (jmiD(R)ijll> = e-im~d~n([3)e-in;' (A.10) where d;m,(/3) = (jmie-i/lJ'ijll> (A.11) The rotation matrices are unitary (A.12) LD~n(R)*D!nll,(R) = b'III' m and satisfy the symmetry relations D~n(R)* = (-l)m-nD~n(R) = D~m(R-1) (A.13) where R - 1 = (-~' + n, [3, -:x + n) is the rotation inverse to R, and the orthonormality relations (" (2" (2,,"". 8n 2 Jo Jo Jo D:"n(:x[3~')* D:"·II·(:x[3·,')sin[3 d[3 d:x d~' = 2j + 1 bjj.bmm ·b,I/J' (A.14) For integer j, j = I, and Il = 0 the rotation matrices reduce to spherical harmonics (A.15) and eq. (A.8) then reduces to the spherical harmollics additioll theorem, (A.16) III where 8 is the angle between the directions specified by 8b ¢1 and 82'¢2' The matrices (A.11) have the symmetry properties d!"n([3) = (-l)m-lId~m([3) = dkm([3) (A.17) and forj = 1,2 are given in Table A.2. 280 Appendix A • Spherical Tensor Formalism Table A.2 The reduced rotation matrices d~" for j = 1,2. dl[ = cos21fJ di o = -(WI2sinfJ dbo = cosfJ dlr = sin21fJ d~2 = cos41fJ dL = 10 + cosfJ)(2cosfJ - 1) dL = -1sinfJ(1 + cosfJ) dio = _(~f2sinfJcosfJ d~o = (~)[ 2sin2fJ dh = 10 - cosfJ)(2cosfJ + 1) dh = -1sinfJ(1 - cosfJ) d~2 = sin41fJ A.2 Clebsch-Gordan Coefficients The vector-addition or Clebsch-Gordan coeffiCients are the coefficients C(i! ;2i~;m,m,m~) = <;, ;,m,m, 1;1;' ;m> (A.18) in the unitary transformation Ijd2jm) = I Ijd2mlm2)Ud2mlm2 Ulj2jm) (A.19) corresponding to the addition, J = J 1 + J 2, of the angular momenta J 1 and J 2' The unitarity implies the relations I C(jljzj;m 1m2m )C(jljzj';m1m 2m') = (jjj,(jmm' (A.20) I C(jljzj;mlmzm)C(jdzj;m~m2m) = (jm[m\(jm2m2 jm and the relation J = J I + J 2 implies C(jljzj3;m 1m Zm 3) = 0, unless m3 = m1 + m2, and l'l(jlj2j3) (A.21) where the triangle condition l'l(jlj2h) means that j3 is one of the numbers (A.22) The phases are chosen in such a way that all the coefficients (A.18) are real and (A.23) where C(jlj2j3;m 1mZ ) == C(jdzj3;m 1,mZ,ml + mz) (A.24) We note the further properties C(jjOj3;m I 0) = (jith (A.25) Sec. A.2 • Clebsch-Gordan Coefficients 281 and C(jjj2j3;OO) = O. unless 2J == jl + j2 + h is even. (A.26) For J integer, i.e.,jl + j2 + j3 even, we have J . J! C(jlhi3;OO) = (-1) - J3 (J _ JJ!-(J---j-z)-!(-J---j-'3-)! x i(2i3 + 1)(j1 + j2 - j3)!(jl - j2 + h)!( -j~+ j2 + j3)!jl 2 L (2J+l)! (A.27) where 2J = jl + jz + j3' The symmetry properties of the Clebsch-Gordan coefficients are most easily expressed in terms of the Wigner 3-j symbol defined by which is invariant under cyclic permutations ofthe columns, and is multiplied by a phase factor under noncyclic ones = = (_ l)a + b + c (a b c) (b c a) (h a '/c) . etc. (A.29) 'X f3~' f3"/ 'X f3 'X and (~ ~ ~) = (_l)a+b+C(a b c) (A.30) 'X f3 7 'X f3 7 gIvmg C(jlj2h;11l1111 2111 3) = (_I)h + h-hC(jlj2j3;ifllifl2ifz 3) 2' 1)1/2 = (_I)h+m2 ( J.3 + C('hhh .. ;11121113-111 1 ) 211+ 1 2' + 1)12 = (_ l)h +",2 ( -'1-=.3'---_ C('"hhh ;1113- 111 2ml-) 211+ 1 = (_I)h -m, (-,21-=>_+_11)1/2 C(j3jlj2;111 3ifz I 111 2) 212 + (A.31) The numerical values of some often occurring Clebsch-Gordan coefficients can be obtained from Table A.3 and eg. (A.31). More extensive tables are available.4 . 5 282 Appendix A • Spherical Tensor Formalism Table A.3 Numerical values of frequently occurring Clebsch-Gordan coefficients; C = C(/t1zl;mtmzm). 1,/ 2 1 1n 11112 m C 1,lzl m}m2m C 1 1 1 000 0 223 1 1 2 0 1 1 1 1 0 1 (W'z 223 2 1 3 (WIZ 1 12 000 (W z 224 000 3(is)12 1 1 2 1 10 (i)'z 224 iTO 2(is)1/2 1 1 2 101 (l:)' Z 224 210 (10)12 1 12 1 1 2 224 10 1 (W z 224 2 11 (i4)' 2 212 000 0 224 202 (1:,:)' 2 212 o 1 1 _(1)12 224 1 1 2 2(W z 2 I 2 101 (i)12 224 2 I 3 (WO z 212 202 (W 2 224 224 1 2 I 2 1 1 2 -(W 2 3 I 3 000 0 2 I 3 000 (W 2 3 1 3 01 1 -(W 2 2 1 3 110 (!)IZ J ! J ! 0 ! ~!~}1.'2 2 I 3 o I I (W Z 3 1 3 1 1 2 z 2 I 3 101 2(is)1 Z -1(W 3 I 3 202 Ct)t2 2 1 3 2T 1 (is)12 3 1 3 2 1 3 I 2 1 3 I 12 (W 2 -2 3 1 3 303 z 2 1 3 202 (t)1 2 1(W 2 I 3 2 I 3 3 1 4 000 2(W Z 3 I 4 1 10 (1:,:)1 Z 222 000 -(W z 3 1 4 101 1(1;)IZ 222 101 _(i4)12 3 1 4 I I (~)I 2 222 I 12 -(W 2 o 3 1 4 2T 1 1(W Z 222 202 (W 2 3 I 4 202 (W 2 223 000 0 3 1 4 1 1 2 1(1;)12 2 223 1 I 1 (W 3 I 4 312 1(W 2 (io)12 223 220 3 1 4 303 (W 2 /2 223 1 0 1 (W 314 2 1 3 1(W 2 (1o)1!2 223 2 I I 3 I 4 3 14 I 223 202 (W/2 The Clebsch-Gordan coefficients also occur in the decomposition rule for the rotation matrices, D!,:l n l(R)D!;2n2(R) = L C(j Ij2j;m 1 m2f1)C(j d2j;n 1 n2v)D{,,(R) jl1\' L C(j d2j;ml m2)C(j Ij 2j ;nln2)D~1 +mpi +1I2(R) (A.32) Sec.